# Question 2931: Diameter Length for a Nonnegative Radius

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0004-02931 · q-20261005-advance-20261006-r05-circle-0056



## Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=5≥0. Find its diameter length.


- G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The circle radius is r=5≥0. Find its diameter length.





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-circle](https://ocme.evemisslab.com/en/mko/mko-circle/)





## Step-by-step derivation

S001 · impliesBefore

orthonormal equal-length units;r=5≥0

After

C={P:distance(P,O)=r} is a boundary,not a filled disk

Conditions: none

The circle is the boundary at distance exactly r from the center,not the filled disk.For r=0 use the degenerate singleton{O},with no inside points and diameter length0.

Step identifiers and dependenciesRule: circle_boundary_convention · Uses: G001





S002 · calculationBefore

diameter length := sum(5, 5)

After

diameter length = 10

Conditions: none

Add two opposite radial lengths to obtain2r;this is a length,not circumference or area.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · impliesBefore

radius=5

After

diameter=10

Conditions: none

The center location does not affect this length;for r=0 the result is0 without division by r.

Step identifiers and dependenciesRule: diameter_twice_radius · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S002 · premise_and_result_audit — Recompute diameter length with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

10

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 20d565a0c5b7005b8e6ebef5b1cb1a5611ab558e63cb83a4b0717c52cf190c23
- candidate record SHA-256: d0bdc4c85fc81e6a67d3908241c27989a739521033e466cf5674582ad03e8396
- semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
- question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r05-circle-0056.json)